Does anybody know what is the biggest $r$ such that $\mathbb{Z}^r$ is isomorphic to a subgroup of $\mathrm{SL}_n(\mathbb{Z})$?

It cannot be bigger that the virtual cohomological dimension of $\mathrm{SL}_n(\mathbb{Z})$, which is $\frac{n(n-1)}{2}$, since the cohomological dimension respects inclusions. But I suspect it must be smaller.